Refinement thresholds by format, against a problem at κ = 1000
At its defaults it draws refinement thresholds by format, against a problem at κ = 1000. A horizontal bar chart of the condition number at which iterative refinement stops working, one bar per floating-point format, with a marked line at the condition number of the problem.
format-ladder is one function in lib/figures/mixed.js —
mixed precision — refinement, and the threshold at 1/u. Everything below came out of it during this build, at
arguments taken from the essays rather than invented for this page. A figure here is the
figure a reader meets in an essay, and if the generator changes, this page changes with it.
At its defaults
Drawn even though every essay passes arguments — which on this site is every essay, at 100% of placements since the standard pass. A default nothing exercises is a trap for the next essay to call this with none, and this is the page where a default that has drifted from the figures around it becomes visible.
A horizontal bar chart of the condition number at which iterative refinement stops working, one bar per floating-point format, with a marked line at the condition number of the problem.
logKappa: 3
The arguments are the ones The other half of a format passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
A horizontal bar chart of the condition number at which iterative refinement stops working, one bar per floating-point format, with a marked line at the condition number of the problem.
logKappa: 4
The arguments are the ones The part of a solver that may be rounded passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
A horizontal bar chart of the condition number at which iterative refinement stops working, one bar per floating-point format, with a marked line at the condition number of the problem.
What it checked while drawing
Every figure above checked its own claims on the way to being drawn, and a claim that failed
would have stopped the picture rather than shipped a wrong one. Those checks used to leave
no trace at all: a passing one returned true and the only evidence the figure had
checked anything was that nothing crashed. The list below is what they actually said, collected
by running this generator with an observer installed — not a description of
what it is believed to check.
16 distinct claims across 3 sets of arguments, grouped below by shape — because most of them are one sentence with a different number in it, and how many separate times that sentence was put to the test is the informative part.
and κ·u is κ times it for fp16 — checked 3 times
fp16's threshold is exactly 1/u — checked 3 times
and κ·u is κ times it for bfloat16
and κ·u is κ times it for tf32
at least one format clears this κ
bfloat16's threshold is exactly 1/u
fp16 and tf32 share a threshold
fp16 has at least bfloat16's threshold
fp32 has at least tf32's threshold
fp64 has at least fp32's threshold
tf32 has at least fp16's threshold
tf32's threshold is exactly 1/u
Against the rule
The rule does not apply to it. It factorises nothing, so there is no residual it could be withholding. That is worth stating rather than leaving blank: a site that reported the rule as satisfied by every generator would be counting mostly generators the rule never reached.
Across the library: the rule bites on 217
of 397 generators —
199 print a residual and
18 are exempt with a published reason;
180 factorise nothing.
Read from lib/residual-rule.js, which is the same body the gate enforces from,
and the gate's last check fails the build if this page and it disagree about any generator.
Where it is called
Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.
The other half of a format
fp16 and tf32 have the same eleven significand bits and their largest numbers are 65,504 and 3.4·10³⁸. For two phases this site simulated the significand alone, so it was obliged to report them as the same format — which is a claim, and a false one.
Methods that were designed apartThe part of a solver that may be rounded
A preconditioner computed and applied with a three-bit significand still returns thirteen correct digits — it costs seventeen extra iterations and nothing else. Round the working arithmetic instead and the step count barely moves while the answer loses exactly the digits the format dropped.